BENEFIT-COST ANALYSIS Financial and Economic Appraisal using Spreadsheets Ch. 3: Decision Rules © Harry Campbell & Richard Brown School of Economics The University of Queensland

Applied Investment Appraisal Conceptualizing an investment as: a net benefit stream over time, or, “cash flow”; giving up some consumption benefits today in anticipation of gaining more in the future. A project as a cash-flow:

Although we use the term “cash flow”, the dollar values used might not be the same as the actual cash amounts. In some instances, actual ‘market prices’ do not reflect the true value of the project’s input or output. In other instances there may be no market price at all. We use the term ‘shadow price’ or ‘accounting price’ when market prices are adjusted to reflect true values.

Three processes in any cash-flow analysis identification valuation comparison

Conventions in Representing Cash Flows Initial or ‘present’ period is always year ‘0’ Year 1 is one year from present year, and so on All amounts accruing during a period are assumed to fall on last day of period

Graphical Representation of Cash Flow Convention Figure 2.4

Comparing Costs and Benefits We cannot compare dollar values that accrue at different points in time To compare costs and benefits over time we use the concept “discounting” The reason is that $1 today is worth more than $1 tomorrow WHY?

Discounting a Net Benefit Stream WHICH PROJECT ? Year0123 Project A Project B

Deriving Discount Factors Discounting is reverse of compounding FV = PV(1 + i) n PV = FV x 1/ (1 + i) n 1/ (1 + i) n is the Discount Factor

Using Discount Factors If i = 10% then year 1 DF 1,10% = 1/(1+0.1) 1 = PV of $50 in year 1 = $50 x = $45.45 What about year 2 and beyond? PV of $40 in year 2 = $40 x x 0.909= $40 x = $33.05 PV = $30 in year 3 = $30 x = $30 x = $22.53

Calculating Net Present Value Net present value (NPV) is found by subtracting the discounted value of project costs from the discounted value of project benefits Once each year’s amount is converted to a discounted present value we simply sum up the values to find net present value (NPV) NPV of Project A = -100(1.0) + 50(0.909) + 40(0.826) + 30(0.751) = -$ = $1.03

Using the NPV Decision Rule for Accept vs. Reject Decisions If NPV ≥ 0, accept project if NPV < 0, reject project

Comparing Net Present Values Once each project’s NPV has been derived we can compare them by the value of their NPVs NPV of A = = $1.03 NPV of B = = $1.99 As NPV(B) > NPV(A) choose B Will NPV(B) always be > NPV(A)? Remember, we used a discount rate of 10% per annum.

Changing the Discount Rate As the discount rate increases, so the discount factor decreases. Remember, when we used a discount rate of 10% per annum the DF was If i = 15% then year 1 DF 1,15% = 1/(1+0.15) 1 = 0.87 This implies that as the discount rate increases, so the NPV decreases. If we keep on increasing the discount rate, eventually the NPV becomes zero. The discount rate at which the NPV = 0 is the “Internal Rate of Return” (IRR).

The NPV Curve and the IRR Where the NPV curve intersects the horizontal axis gives the project IRR Figure 2.5:

The IRR Decision Rule Once we know the IRR of a project, we can compare this with the cost of borrowing funds to finance the project. If the IRR= 15% and the cost of borrowing to finance the project is, say, 10%, then the project is worthwhile. If we denote the cost of financing the project as ‘r’, then the decision rule is: If IRR ≥ r, then accept the project If IRR < r, then reject the project

NPV vs. IRR Decision Rule With straightforward accept vs. reject decisions, the NPV and IRR will always give identical decisions. WHY? If IRR ≥ r, then it follows that the NPV will be > 0 at discount rate ‘r’ If IRR < r, then it follows that the NPV will be < 0 at discount rate ‘r’

Graphical Representation of NPV and IRR Decision Rule Figure 3.0

Using NPV and IRR Decision Rule to Compare/Rank Projects Example 3.7: IRR vs. NPV decision rule IRRNPV 0123 (at 10%) A %$181 B %$137 If we have to choose between A and B which one is best? NPV and IRR give identical results for accept vs. reject decisions when considering an individual project. IRR should not be used to rank the projects.

Switching and Ranking Reversal NPVs are equal at 15% discount rate At values of r < 15%, A is preferred At values of r > 15%, B is preferred Therefore, it is safer to use NPV rule when comparing or ranking projects. Figure 3.1

Choosing Between Mutually Exclusive Projects In example 3.8, you need to assume the cost of capital is: (i) 4%, and then, (ii) 10% IRR (A) > IRR (B) At 4%, NPV(A) < NPV (B) At 10%, NPV(A) > NPV (B)

Other Problems With IRR Rule Multiple solutions (see figure 2.8) No solution (See figure 2.9) Further reason to prefer NPV decision rule. Figure 2.8 Multiple IRRs

Figure 2.9 No IRR

Using Annuity Factors Year01234 NCF If i=10%, NPV= (DF 1,10% )+45(DF 2,10% )+35(AF 4,10% - AF 2,10% )= (0.909)+45(0.826)+35( )=32.81 In general if you want to find the AF for years X to Y, then use the following rule (where Y>X): AF x to y =AF y -AF x-1 Year01234 NCF If i=10%, NPV=100+35(DF 1,10% )+35(DF 2,10% )+35(DF 3,10% )+35(DF 4,10% )= = (DF 1,10% +DF 2,10% +DF 3,10% +DF 4,10% )= = (AF 4,10% )= (3.17)=10.95

Using Annuity Factors Annuity Tables could be also used to find IRR. Remember, IRR is the rate at which NPV=0. NPV = (AF 3,x% )=0 AF 3,x% =2362/1000=2.362 From Annuity Tables, see year 3 and look for the column in which AF 3,x% = This entry appears in the column corresponding to a discount rate of 13%. Therefore, IRR=13%. Year0123 NCF

Benefit-Cost Ratio Decision-Rule

However, when it comes to comparing or ranking two or more projects, again assuming no budget constraint*, the BCR decision-rule can give incorrect results. *If there is a budget constraint, BCR should not be used. PV BenefitsPV CostsNPVBCR Project A$100$60$ Project B$80$45$351.78

Profitability ratio or Net Benefit Investment Ratio (NBIR) Decision Rule

Problems With NPV Rule Capital rationing – Use Profitability Ratio (or Net Benefit Investment Ratio Indivisible or ‘lumpy’ projects – Compare combinations to maximize NPV Projects with different lives – Renew projects until they have common lives: LCM – Use Annual Equivalent method

Under capital rationing ProjectPV(K)PV(B)NPV(rank)B/K(rank) A (5)1.3 (2) B (4)1.08 (5) C (1)1.52 (1) D (2)1.24 (3) E (3)1.11 (4) Limited Budget = 800$, and if projects are divisible then: Using NPV method → C+D+0.4E → NPV total =$103+$94+0.4*$58=$220 Using profitability ratio (NBIR) → C+A+D+0.2E → NPV total =$103+$30+$94+0.2*$58=$239 Table 3.3 Ranking of Projects by NPV and Profitability Ratio ($thousand)

Lumpy projects ProjectPV(K)PV(B)NPV (rank)B/K (rank) A (4)1.44 (2) B (3)1.42 (3) C (1)1.52 (1) D (2)1.24 (3) Table 3.4 Ranking Lumpy Projects ($ thousands) Limited Budget=$300, and if projects are indivisible then: Using NBIR method → C → NPV total =$104 Try to use the full budget → A+B → NPV total =$55+$73.5=$128.5

In situations of capital rationing where funds are constrained and potential investments must be ranked, we should use the profitability ratio (NBIR) rather than the absolute size of the NPV. However, when investments cannot be divided into smaller parts (lumpy investments), we should give consideration to whether or not our selection utilizes the full budget available; and, if not, to whether accepting a combination of lower ranking, smaller projects would generate a higher overall NPV.

Projects with Different Lives It is possible to convert any given amount, or any cash flow, into an annuity. ProjectInitial CostAnnual Net costsLife (Years) A$40,000$2,8004 B$28,000$4,4003 PV of Costs (A) = - $48,876 PV of Costs (B) = - $38,956 Using the Annual Equivalent Cost (or Benefit) Method (AF 4,10% =3.17, AF 3,10% =2.49): AEC (A) = $48,876/3.17 = $15,418 AEC (B) = $38,956/2.49 = $15,645 AEC (B)> AEC (A), therefore, choose A.

Projects with Different Lives Establishing Equal Project Lives (use LCM) Project A B Using a 10% discount rate: PV of costs (A) = (0.683)-40(0.467)-2.8(6.814)= PV of costs (B) = (0.751)-28(0.564)-28(0.424)-4.4(6.814)= = Project A is less costly than B, choose A.